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Two-Way ANOVA Tutorial
Purpose of Two-Way ANOVAFactorial DesignsTypes of Two-Way ANOVAsExamining Means TablesExamining Graphs of MeansDividing the VarianceComparing the Variance
PURPOSE OF TWO-WAY ANOVA
You have just learned about a One-Way ANOVA, in which we determine whether the
effect of an independent variable is significant. In a Two-Way ANOVA, we determine
whether there are significant main effects of each of two independent variables, and we
also determine whether there is a significant interaction between two independent
Frame 2A main effect is the overall effect of one independent variable. Suppose I have done anexperiment manipulating both the type of therapy (Psychotherapy or Prozac) depressedpatients receive, and the setting in which the therapy is given (inpatient or outpatient).
The main effect of the type of therapy refers to the differences between the differenttherapies, ignoring whether they were given in an inpatient or outpatient setting. Themain effect of setting refers to the difference between the inpatient and outpatientsettings, ignoring which type of therapy was given.
Frame 3An interaction means that the effect of one independent variable changes depending onthe level of the other independent variable. Suppose that Prozac works better thanPsychotherapy for inpatients, but Psychotherapy works better than Prozac foroutpatients. That would be an interaction!
An interaction would also occur if the effect of the type of therapy was greater in onesetting than in the other. For example, there might be a small difference ineffectiveness between Prozac and Psychotherapy for inpatients, but a large differencefor outpatients.
Frame 4You might wonder why we even bother with Two-Way ANOVA, which is obviously morecomplicated than One-Way ANOVA. Why not just do a One-Way ANOVA to test the
effect of the type of therapy, and do another One-Way ANOVA to test the effect of thesetting?
The answer is that the separate One-Way ANOVAs would not let us test the interaction.
It is important for researchers to find out if interactions exist. If the most effectivetreatment for depression is different depending on whether the setting is inpatient oroutpatient, people need to know that!
Another advantage of a Two-Way ANOVA is that the amount of non-systematicvariance will be reduced. This is a good thing, because it means that we can accountfor more of the differences in people’s scores. The reason that we can reduce theamount of non-systematic variance is that some of the differences in scores would becounted as non-systematic in a One-Way ANOVA, but those differences might really becaused by the second independent variable or by the interaction in a Two-Way ANOVA.
Two-Way ANOVAs are used when the researcher has conducted a factorial design, a
research method in which all combinations of levels of two or more independent
variables are measured.
A level just means a value of an independent variable. If my independent variable isTemperature, my levels might be Warm and Cold (I can have more than two levels too).
The levels of the independent variable Noise Level might be Quiet and Noisy.
The combinations of levels of independent variables are called conditions. For theabove example, I would have to measure people in four conditions: Quiet-Warm, Quiet-Cold, Noisy-Warm, Noisy-Cold.
Also keep in mind that we sometimes use the term factor to refer to an independentvariable.
Frame 2We use a fairly simple notational system to indicate how many independent variablesare in the factorial design, and how many levels each independent variable has.
The notation 2x2 means that there are two independent variables, each with two levels,as in the example on the previous page. Notice that if you multiply the two numbers, ittells you how many conditions you will need to measure.
If we added a third level of Temperature, so that the levels were Cold, Warm, and Hot,and we still had two levels of Noise Level (Quiet and Noisy), we would have a 2x3factorial. We could also call it a 3x2 factorial.
If I add a third independent variable to my factorial, I will have to use another number inthe notation. For example, suppose I add Light Level (Dim and Bright) as a third factor.
Now I have a 2x3x2 design – two levels of Noise Level, three levels of Temperature,and two levels of Light Level. Notice that I would have to measure people in 12conditions for that design! Yikes!
TYPES OF TWO-WAY ANOVAS
There are three types of Two-Way ANOVAs. The particular ANOVA you use will depend
on how the data were collected in your factorial design.
In a between subjects factorial, different individuals are measured in each condition. ATwo-Way Between Subjects ANOVA would be appropriate (you will actually learn howto compute this particular version).
In a within subjects factorial, the same individuals are measured in each condition. Ifthere are six conditions, each individual would have to be measured six times. A Two-Way Within Subjects ANOVA would be appropriate. This is also sometimes called aTwo-Way Repeated Measures ANOVA.
Frame 2The third type of factorial is the mixed factorial. This means that one independentvariable is between subjects and the other independent variable is within subjects. Forexample, I might measure one group of people in a Quiet condition and another groupin a Noisy condition. This makes Noise Level a between subjects factor. I mightmeasure everybody in both Warm and Cold conditions; this would make Temperature awithin subjects factor. A Two-Way Mixed ANOVA would be appropriate for this type offactorial design.
EXAMINING MEANS TABLES
When you interpret the results of a factorial design, it is helpful to examine the means of
the conditions in addition to the results of your Two-Way ANOVA. There are two ways
to examine the pattern of means: by looking at a table of means, or by examining a
graph of means.
Examining a table or graph of means will help you understand the pattern of maineffects and interaction. Looking just at the means will NOT tell you whether any ofthese effects are significant; the F-tests in the ANOVA are needed for that.
Frame 2Below is an example of a table of hypothetical means for a 2x3 factorial.
Frame 3To examine the main effects in a table of means, you should first determine themarginal means, which are the means of the levels of each factor. You can write thesemeans on the margins of the table, as shown below:
Frame 4Notice that the marginal mean for the Quiet conditions is 30, compared to 50 for theNoisy conditions. When the marginal means are different for the levels of a factor, itsuggests a main effect of that factor. So, this looks like a main effect of Noise Level.
(But remember that we don’t know if the main effect is significant until we actually do theANOVA).
You’ll also notice that the marginal means for the levels of Temperature are different,suggesting a main effect of Temperature. When a factor has more than two levels, ANY
difference in marginal means suggests that there is a main effect. For example, therewould still be a main effect even if the marginal means had been 30, 30, and 50 insteadof 30, 40, and 50.
Frame 5We should also examine the pattern of means for an interaction. To do this, we mustfirst find the simple effects, which are the effects of a factor for each level of anotherfactor.
Let’s look at the simple effects of Noise Level for each level of Temperature (you willwant to go back to the table of means to see this). In Cold rooms, there is a 20 pointsimple effect of Noise Level (the Noisy mean is 20 points higher than the Quiet mean).
In Warm rooms, there is a 20 point simple effect of Temperature, and the same is truefor Hot rooms.
If the simple effects are the same for each level of the other factor, there is NOT aninteraction. Therefore, this pattern of means suggest that there is not an interaction ofNoise Level and Temperature.
Frame 6Let’s look at another example of a table of means.
In this example, you will notice that the simple effect of Noise Level changes across thedifferent levels of Temperature. The simple effect is 20 points in the Cold room andWarm room, but 0 points in the Hot room. This suggests that there is an interactionbetween Noise Level and Temperature. There is an interaction if the simple effectschange in amount or difference, direction of difference, or both.
EXAMINING GRAPHS OF MEANS
The other way to examine the pattern of means is to look at a graph. You will probably
find it easier to determine if there is an interaction by looking at the graph. Here is an
example of a graph showing the means of a factorial design:
Frame 2Notice that the graph is made so that the mean of each condition is indicated by a circleon the graph. This is a 2x3 factorial, so there are six conditions. Notice also that one ofthe factors (Temperature) is indicated on the x-axis, and the levels of the other factor(Noise) are shown by using different lines. Mean scores on the dependent variable areindicated on the y-axis. Normally, the units of measurement would be marked off on they-axis.
If the lines one the graph are parallel to each other, that means the simple effects arenot changing; therefore there is NOT an interaction. If the lines are not parallel at anypoint, it suggests that there IS an interaction. In this example, the lines are parallel (thespace between them stays the same), so there is not an interaction.
Frame 3You can also tell whether there are main effects in the pattern of means by looking atthe graph, but it is easier to do this by looking at a table of means. When looking at agraph, you have to visually estimate the marginal means. I have placed black trianglesto indicate my estimates of the marginal means for cold, warm, and hot, and redtriangles to indicate my estimates for the marginal means for quiet and noisy.
Frame 4When you want to determine from a graph if there are main effects, estimate whereeach marginal mean is, and make some kind of mark, as I did with the triangles on thelast page. The key thing to remember is that only the height on the graph matters,because the mean scores on the dependent variable are graphed on the y-axis.
If the marginal means for a factor are at different heights on the graph, it means thatthere is a main effect for that factor. From the graph on the last page, we can see thatthe marginal mean for hot is higher than for cold or warm: therefore, there is a maineffect of Temperature. We can also see that the marginal mean for quiet (the upper redtriangle) is higher than the marginal mean for noisy (the lower red triangle). So there isalso a main effect of Noise Level.
Frame 5Here’s another example. What effects do you think are present?
Frame 6If you said that there is an interaction and a main effect of the Setting, you’re right!Clearly, the lines are not parallel, so there is an interaction. The marginal mean for theoutpatient setting is higher than the marginal mean for inpatients, so there is a maineffect of setting. The marginal means for Therapy A and B are at the same height, sothere is not a main effect of Setting.
DIVIDING THE VARIANCE
So far we have been focused on the pattern of means. Remember that we have to
actually do the ANOVA to find out whether each effect (the two main effects and the
interaction) is statistically significant.
Remember also that ANOVA is a two-step process: first we divide the variance intodifferent parts, then we compare the parts of the variance.
In the One-Way ANOVA, we divided the variance into Between Groups and WithinGroups. We do the same thing for a Two-Way ANOVA, but we also take it a stepfurther by dividing the Between Groups variance up into three separate parts.
Frame 2Suppose we call our two independent variables Factor A and Factor B. The totalvariance will be divided into:
Factor A varianceFactor B varianceAxB interaction varianceWithin Groups variance
The first three parts of the variance together make up the Between Groups variance.
Each of these types of variance is influenced by the effect it is named for (Factor Avariance is influenced by the main effect of Factor A), which is a systematic influence.
Each type of variance is also influenced by non-systematic things (measurement errorand individual differences). The Within Groups variance, just as in the One-WayANOVA, has only non-systematic influences.
COMPARING THE VARIANCE
Now that we have divided the variance into parts, the second step is to compare the
parts of the variance. You will recall that we did this in the One-Way ANOVA by
computing a ratio called the F-test. We do the same thing here, only now we get to do
THREE F-tests: one for each main effect, and one for the interaction.
Remember that in ANOVA, we use the term “Mean Square” to refer to variance.
Frame 2The F-test for the main effect of Factor A is:
The F-test for the main effect of Factor B is:
Frame 3Just as before, the larger the F we get, the more likely it is that the effect is statisticallysignificant. We expect the F to be about 1.0 if there is no effect at all, because in thatsituation the F would simply be dividing non-systematic influences by non-systematicinfluences.
Each F-test is a separate question. In other words, the fact that one of the three F-testsis significant does not tell us anything about whether the other two F-tests are significantor not.
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